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An asymptotic Expansion of the Double Gamma Function

Chelo Ferreira

1

and Jos´ e L. L´ opez

2

1Departamento de Mat´ematica Aplicada, Facultad de Ciencias, Universidad de Zaragoza, 50013-Zaragoza, Spain. e-mail: cferrei@posta.unizar.es.

2 Departamento de Mat´ematica e Inform´atica,

Universidad P´ublica de Navarra, 31006-Pamplona, Spain. e-mail: jl.lopez@unavarra.es.

ABSTRACT

The Barnes double Gamma functionG(z)is considered for large argumentz. A new integral representation is obtained for log G(z). An asymptotic expansion in decreasing powers of z and uniformly valid for

|Argz| < πis derived from this integral. The expansion is accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is very accurate for realz. The accuracy of the error bound decreases for increasing Argz.

1991 Mathematics Subject Classification: 41A60 33E05

Keywords & Phrases: Barnes double Gamma function, asymptotic expansions.

1. Introduction

The double and generalized double gamma functions were introduced and primarily investigated by Barnes [2]-[4], who applied this functions to the theory of elliptic and theta functions. Some other mathematical applications of the double gamma function may be found in [9]-[11] and [20]. The double gamma function is used in [20] to prove the classical Kronecker limit formula. Its p-adic analytic extension appeared in a formula of Cassou-Nogu´es for the p-adic L functions at the origin [7]. On the other hand, the theory of the double gamma function is used in [9]-[11] and [21] to evaluate some series involving the zeta function.

A second field of applications is found in the study of determinants of Laplacians. In fact, multiple gamma functions evaluated at 1/2 may be expressed in terms of the functional determinant of Laplacians of the n-sphere, which have been a recent subject of research due to their relevance to Superstring Theory [23]. Toeplitz determinants with special rational generating functions may be evaluated in terms of the double gamma function and the Gauss hypergeometric function [5]. Several properties of the Gamma and double gamma functions may be deduced from the application of the zeta regularization to the determinants of certain operators [24]. The double gamma function plays a key role in deriving the determinant of the Laplacian on spinor fields on a Riemann surface in terms of the value of Selberg zeta function at the middle of the critical strip [19]. Some other applications of the double gamma function to the study of determinants of Laplacians may be found in [8],[16] and [18].

(2)

The double gamma function G(z) is the integral function defined by the infinite product [2]:

G(z + 1) = (2π)z/2e−[(1+γ)z2+z]/2

Y

k=1

 1 + z

k

k

e−z+z2/2k



, (1)

where γ is the Euler-Mascheroni constant. It satisfies the recursion relation G(z + 1) = Γ(z)G(z) and, for integer positive z, it verifies G(1) = G(2) = 1 and G(m) = 1!2!...(m − 3)!(m − 2)! for m ≥ 2.

Although useful in many applications, the double gamma function G(z) has not appeared in the tables of the most well-know special functions, and is just cited in an exercise proposed by Whitaker and Watson [[25], p. 264] and used by Gradshteyn and Ryzhik [[12], p. 661, eq. 6.441(2); p. 937, eq. 8.333].

However, motivated by its recent increasing interest, Richard Askey proposed recently, in the ”panel discussion” of the San Diego symposium on asymptotics and applied analysis (January, 2000), a deeper study of this function. In particular, the derivation of asymptotic expansions of G(z) is necessary for approximating this function at large values of z.

Asymptotic expansions of some functions that generalize the double gamma function have been ob- tained by Matsumoto [13], [14] and Billingham and King [6]. Matsumoto studies, among others, the function Γ2(z, (1, w)) introduced by Barnes in [4] as a generalization of the double gamma function:

G(z) = (2π)z/2Γ−12 (z, (1, 1)). Billingham and King study the generalized double gamma function ¯G(z, τ ), which satisfies the generalized recursion relation ¯G(z + 1, τ ) = Γ(z/τ ) ¯G(z) and the normalization condi- tion ¯G(1, τ ) = 1. It was introduced by Barnes in [3] as a different generalization of the double gamma function: G(z) = ¯G(z, 1).

A complete asymptotic expansion of log Γ2(z, (1, w)) is obtained in [13] and [14] from an integral representation of this function. But this is an expansion in decreasing powers of w and therefore, the asymptotic expansion of G(z) in decreasing powers of z can not be obtained from Matsumoto’s expansion.

The first terms of the asymptotic expansions of the generalized double gamma function ¯G(z, τ ) for large or small z or τ have been obtained in [6] by using the method of matched asymptotic expansions to solve the difference equation. The first terms of uniform asymptotic expansions are also obtained there.

Therefore, an asymptotic approximation of log G(z) = log ¯G(z, 1) for small or large z may be obtained from the asymptotic approximation of log ¯G(z, τ ) for small or large z respectively and fixed τ = 1.

Nevertheless, for small z, several complete convergent expansions of log G(z) in powers of z are given in [[9], eqs. (2.1), (2.15), (2.25)]. On the other hand, complete asymptotic expansions of log G(z) for large z are not known. The purpose of this paper is to obtain a complete asymptotic expansion of log G(z) for large z with error bounds.

In section 2, we obtain an integral representation of log G(z) from which we derive a complete asymp- totic expansion of log G(z) for large z. In section 3 we use the error test and Cauchy’s formula for obtaining accurate error bounds at any order of the approximation. Numerical examples are shown as an illustration. A brief summary and a few comments are given in section 4.

2. Asymptotic expansion

The starting point for deriving an asymptotic expansion of log G(z) is a suitable integral representa- tion. It may be obtained from [[9], eq. (2.15)],

X

k=2

(−1)k ζ(k)

k(k + 1)zk+1=(log(2π) − 1)z

2 + (γ − 1)z2 2 +

z log Γ(z + 1) − log G(z + 1), |z| < 1.

(2)

(3)

Introducing the integral representation of the zeta function [[1], eq. 23.2.7] into the left hand side of the above equation, interchanging sum and integral and after trivial manipulations we obtain

log G(z + 1) = (log(2π) − 1)z

2 + (γ − 1)z2

2 + z log Γ(z + 1) + I(z), |z| < 1, (3) where I(z) is the integral

I(z) ≡ Z

0

e−zx− 1 + xz − (xz)2/2 x2(ex− 1) dx.

This integral defines an analytic function of z for Re(z) > −1 [[22], p. 30, theorem 2.3]. Therefore, the right hand side of (3) defines the analytic continuation of log G(z + 1) to Re(z) > −1.

The asymptotic expansion of I(z) for large z then provides an asymptotic expansion of log G(z + 1).

For obtaining an asymptotic expansion of I(z) we divide the integral into two pieces:

I(z) = I1(z) + I2(z), where

I1(z) ≡ Z

0

 −1 + xz − (xz)2/2

x2(ex− 1) +e−zx

x3 −e−zx

2x2 +e−zx 12x

 dx,

I2(z) ≡ Z

0

e−zx x3

"

x ex− 1−

2

X

k=0

Bk

k! xk

# dx,

and Bkare the Bernoulli numbers. The integral I1(z) may be evaluated by means of elementary techniques by writing

I1(z) = lim

→0



− Z



dx

x2(ex− 1)+ z Z



dx

x(ex− 1) −z2 2

Z



dx (ex− 1)+ Z



e−zx x3 dx −1

2 Z



e−zx x2 dx + 1

12 Z



e−zx x dx

 .

(4)

Integrating by parts we have, for  → 0, Z



e−zx

x dx = − log  − log z − γ + O(), Z



e−zx x2 dx = 1

+ z(log  + log z + γ − 1) + O(), Z



e−zx

x3 dx = 1 22 −z

 +z2 2

 3

2 − log  − log z − γ



+ O().

On the other hand, with the change of variable x = log y, Z



dx

ex− 1 = − log  + O().

In order to calculate the two remaining integrals in (4) we separate their integrand in two terms:

1 ex− 1 ≡ 1

x+1 − ex+ x x(ex− 1)

and expand 1 − ex+ x in power series of x. Using [[1], eq. 23.2.7] we obtain Z



dx

x(ex− 1) = 1

+1 2log  −

X

n=2

ζ(n)

n(n + 1)+ O(),

(4)

Z



dx

x2(ex− 1) = 1 22 − 1

2− 1

12log  +1 2

X

n=2

ζ(n) n(n + 1)−

X

n=2

ζ(n)

n(n + 1)(n + 2)+ O().

Taking the limit z → −1 in (2),

X

n=2

ζ(n) n(n + 1) =1

2(log(2π) − γ) . Introducing these calculations in (4) we find

I1(z) = 3 4 −γ

2

 z2+1

2(1 − log 2π)z −1 2



z2+ z +1 6



log z + C, (5)

were the constant C is defined by

C ≡γ

6 −log(2π)

4 +

X

n=2

ζ(n)

n(n + 1)(n + 2). (6)

Integrating (2) with respect to z and taking again the limit z → −1 we have

X

n=2

ζ(n)

n(n + 1)(n + 2)= log(2π)

4 −γ

6 − 1 12−

Z 0

−1

t log Γ(t + 1)dt + Z 0

−1

log G(t + 1)dt.

Using now [[1], eqs. 6.1.3 and 6.1.41], the definition (1) of G(z) and [[17], p. 647, eq. 1], we obtain C = − log A, where A is Glaisher’s constant defined by

log A ≡ lim

n→∞

( log

n

Y

k=1

kk

!

− n2 2 +n

2 + 1 12



log n +n2 4

)

. (7)

A more stable numerical algorithm useful for evaluating log A is given by (6). Another one can be obtained using by using the Euler-Maclaurin formula [[26], p. 36] on the right hand side of (7),

log A =1 4 + 1

12 Z

1

B4− B4(x − bxc)

x3 dx = 1

4+ 1

12

X

n=1



6n2+ 6n + 1 log

 n

n + 1



+ 6n + 3



' 0.2487544770337843.

On the one hand, the function I1(z) given in (5) is analytic in |Arg(z)| < π. On the other hand, by using the Cauchy residua theorem we obtain that the integral I2(z) may also be written as

I2(z) = Z ∞e

0

e−zx x3

"

x ex− 1 −

2

X

k=0

Bk k!xk

#

dx, (8)

where ϕ is any angle verifying |ϕ| < π/2 and −π/2 − ϕ ≤Arg(z) ≤ π/2 − ϕ. Therefore, the right hand side of (5) plus the right hand side of (8) define the analytic continuation of log G(z + 1) to the sector

|Arg(z)| < π if we define I2(z) by (8) with ϕ verifying −π/2 − ϕ ≤Arg(z) ≤ π/2 − ϕ.

Once we have calculated I1(z) exactly, I2(z) must be approximated asymptotically. For that purpose we substitute x/(ex− 1) in this integral by the expansion [[1], eq. 23.1.1],

x ex− 1 =

N −1

X

n=0

Bn

n!xn+ rN(x), N = 1, 2, 3, 5, 7, 9, ..., (9)

(5)

where x 6= ±2mπi, m ∈ N| and rN(x) = O(xN) as x → 0, obtaining

I2(z) =

N −1

X

n=1

B2n+2

2n(2n + 1)(2n + 2)z2n + RN(z), N = 1, 2, 3, ..., (10) where

RN(z) = Z ∞e

0

e−zx

x3 r2N+2(x)dx. (11)

Therefore, plugging the expansion of I2(z) and the value of I1(z) into I(z) and this into equation (3), we obtain the following theorem.

Theorem 1. For |Arg(z)| < π, the logarithm of the double gamma function admits the expansion

log G(z + 1) =1

4z2+ z log Γ(z + 1) − 1 2z2+1

2z + 1 12



log z − log A+

+

N −1

X

n=1

B2n+2

2n(2n + 1)(2n + 2)z2n + RN(z), N = 1, 2, 3, ...,

(12)

In the following section we obtain bounds for RN(z) which show that in fact, (12) defines an asymptotic expansion of log G(z + 1).

3. Error bounds

We derive now error bounds for RN(z) at any order N of the approximation (12). We obtain first error bounds for rN(x) in three different regions of the variable x: 0 ≤ x < 2π, |x| < 2π and the shaded region depicted in figure 1. Then, we introduce these bounds into the integral (11) defining RN(z).

3.1. Bounds for rN(x)

For 0 ≤ x < 2π, equation (9) defines a convergent expansion and therefore, for N = 1, 2, 3, ..., r2N+2(x) = r(1)N (x) + rN(2)(x),

where

r(1)N (x) ≡

X

n=0

B4n+2N+2

(4n + 2N + 2)!x4n+2N+2 and

r(2)N (x) ≡

X

n=0

B4n+2N+4

(4n + 2N + 4)!x4n+2N+4.

For real x < 2π, all the terms in the sum defining r(1)N (x) are negative for odd N and positive for even N , whereas the terms in the sum defining r(2)N (x) are all positive for odd N and negative for even N . Using the bounds for the Bernoulli numbers given in [[1], eq. 23.1.15] we find

|r(2)N (x)| ≤2

X

n=0

 x 2π

4n+2N+4 1

1 − 2−4n−2N−3 ≤ 2(x/2π)2N+4

1 − (x/2π)4 + 2(x/2π)2N+4 (22N+3− 1)(1 − (x/4π)4)

≤2(x/2π)2N+2

1 − (x/2π)4 ≤ |rN(1)(x)|.

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Im(w)

Re(w)

x r

r 2 I I i__

-2 I I i__

r C

Figure 1. The circle of radiusr ≡ π cos ϕcentered atξ, withξ ∈ (0, x), used in the Cauchy definition ofr2N+2(x) must be contained in the shaded region. This region is defined by the set{w ∈ C/,|w − x| < π cos ϕ,0 ≤ |x| < ∞}.

The third inequality above may be derived by observing that the inequality



1 + 22N+3− 1 16



a4+ a2− 22N+3+ 1 ≤ 0

holds for 0 ≤ a ≡ x/2π < 1 and N = 1, 2, 3, .... Therefore we find that sign(r2N+2(x)) = (−1)N for 0 ≤ x < 2π and N = 1, 2, 3, ... and two consecutive error terms in the expansion (9) have opposite sign.

Then, applying the error test (see for example [[15], p. 68] or [[26], p. 38]) we find,

|r2N+2(x)| ≤ |B2N+2|

(2N + 2)!x2N+2, 0 ≤ x < 2π, N = 1, 2, 3, ... (13) On the other hand, for complex x with |x| < 2π we have, using [[1], eq. 23.1.15],

|r2N+2(x)| ≤

X

n=N

|B2n+2|

(2n + 2)!|x|2n+2≤ 2

x 2π

2N+2 1

1 − |x/2π|+(22N+1− 1)−1 1 − |x/4π|

 .

(14)

Finally, for any x ∈ C/we consider the explicit expression for r2N+2(x) given by the Lagrange form for the remainder of the Taylor expansion (9):

r2N+2(x) = 1 (2N + 2)!

d2N+2 dx2N+2

 x

ex− 1

 x=ξ

x2N+2, N = 1, 2, 3, ...,

where ξ ∈ (0, x). By the Cauchy theorem, r2N+2(x) = 1

2πi Z

C

wdw

(w − ξ)2N+3(ew− 1)x2N+2, N = 1, 2, 3, ...,

(7)

where C is a circle with center at the point ξ that does not enclose singularities of (ew− 1)−1. In order to find bounds of RN(z) we require bounds of r2N+2(x) valid for fixed Arg(x) = ϕ with |ϕ| < π/2 and 0 ≤ |x| < ∞. Therefore, we make the change of variable w = ξ + π cos ϕe, (observe that π cos ϕ < the distance of the x−axis to the first singularities ±2πi of (ew− 1)−1, see figure 1), obtaining

|r2N+2(x)| ≤ C(ϕ) x2N+2

(π cos ϕ)2N+2, N = 1, 2, 3, ..., (15) where C(ϕ) is a bound of |w/(ew− 1)| in the shaded region depicted in figure 1.

The maximum of the function |w/(ew− 1)| in the shaded region of figure 1 is situated at a tangent point of the level lines of that function with the curve w(x) ≡ x + iy(x) limiting that region,

y(x) =





x tan ϕ + π if −π sin ϕ cos ϕ ≤ x < ∞, pπ2cos2ϕ − x2 if −π cos ϕ ≤ x < −π sin ϕ cos ϕ,

−p

π2cos2ϕ − x2 if −π cos ϕ ≤ x < π sin ϕ cos ϕ, x tan ϕ − π if π sin ϕ cos ϕ ≤ x < ∞.

Therefore, we can take the constant C(ϕ):

C(ϕ) = Maxx0,x1

( px20+ π2cos2ϕ ± 2x0π sin ϕ cos ϕ sin(x0tan ϕ) ,

π cos ϕ r

e2x1+ 1 − 2ex1cos

2cos2(ϕ) − x21





 ,

(16)

where x0 are the solutions of the equation tan ϕ sin(x tan ϕ) = ex + cos(x tan ϕ) in the interval [−π| sin ϕ| cos ϕ, ∞) and x1 are the solutions of the equations

x sinp

π2cos2ϕ − x2= ±(ex− cosp

π2cos2ϕ − x2)p

π2cos2ϕ − x2 in the respective intervals [−π cos ϕ, ∓π sin ϕ cos ϕ).

3.2. Bounds for RN(z)

For |Arg(z)| < π/2 we can take ϕ = 0 in (11) and therefore, we can plug the bounds (13) and (15) into the integral (11) defining RN(z). We can use (13) for 0 ≤ x < 2π and (15) for x ≥ 2π with C(0) = π/(1 − e−π), obtaining

|RN(z)| ≤ |B2N+2| (2N + 2)!

Z 0

e−xRe(z)x2N−1dx+

Z

e−xRe(z)x2N−1

 1

(1 − e−π2N+1 − |B2N+2| (2N + 2)!



dx, N = 1, 2, 3, ...

Therefore, we obtaining the following theorem,

Theorem 2. For |Arg(z)| < π/2, an error bound for the remainder RN(z) in the expansion (12) of the logarithm of the double gamma function is given by

|RN(z)| ≤ CN(z)

(Re(z))2N, N = 1, 2, 3, ... (17)

(8)

where CN(z) is given by

CN(z) = |B2N+2|

2N (2N + 1)(2N + 2)+ e−2πRe(z)

 1

(1 − e−π2N+1

|B2N+2| (2N + 2)!

2N −1 X

k=0

 2N − 1 k



k!(2πRe(z))2N−k−1.

The bound (17) is not satisfactory for Arg(z) close to ±π/2 and not valid for π/2 ≤ |Arg(z)| < π. A more accurate error bound may be obtained for Arg(z) close to ±π/2 which is valid also in the whole sector |Arg(z)| < π by using the freedom we have when choosing the parameter ϕ in (11). For that purpose we introduce in (11) the bound (14) for |x| ≤ π and the bound (15) for |x| > π. After trivial manipulations and choosing ϕ = −12Arg(z), we find the following theorem

Theorem 3. For |Arg(z)| < π, an error bound for the remainder RN(z) in the expansion (12) of the logarithm of the double gamma function is given by

|RN(z)| ≤ CN(z)

|z cos(Arg(z)/2)|2N, N = 1, 2, 3, ... (18) where

CN(z) ≡4(2N − 1)!

(2π)2N+2



1 + 2

3(22N+1− 1)

 +

 C(−Arg(z)/2)

(π cos(Arg(z)/2))2N+2 − 4 (2π)2N+2



1 + 2

3(22N+1− 1)



×

e−π|z cos(Arg(z)/2)|

2N −1

X

k=0

 2N − 1 k

 k!



π|z| cos Arg(z) 2

2N−1−k

and C(ϕ) is given in (16).

This bound shows that in fact, expansion (12) is an asymptotic expansion of log G(z + 1) in the sector

|Arg(z)| < π.

Tables 1-4 show numerical experiments about the approximation supplied by expansion (12) for some values of z and the accuracy of the error bounds (17) and (18).

Table 1 (Arg(z) = 0)

First Relative Relative Second Relative Relative

|z| I2(z) order approx. error error bound order approx. error error bound 1 -0.0012455229 -0.0013888888 0.115 2.24 -0.0011904761 0.0442 13.4 2 -0.0003360773 -0.0003472222 0.0332 0.0425 -0.0003348214 0.00374 0.032 5 -0.0000552441 -0.0000555555 0.00564 0.00574 -0.0000552380 0.000110 0.000115 10 -0.0000138691 -0.0000138888 0.00142 0.00143 -0.0000138690 7.009e-6 7.18e-6 20 -3.4709876e-6 -3.4722222e-6 0.0003568 0.0003572 -3.4709821e-6 4.45e-7 4.47e-7 50 -5.55552382e-7 -5.5555555e-7 5.713e-5 5.715e-5 -5.55552381e-7 1.142e-8 1.143e-8 100 -1.38886905e-7 -1.38888888e-7 1.42852e-5 1.42859e-5 -1.38886905e-7 7.142e-10 7.143e-10

Figure 2. Second, third and sixth columns represent the integral I2(z) = log G(z+1)−14z2−z log Γ(z+

1) + 12z2+12z +121 log z + log A, approximation (10) for N = 2 and approximation (10) for N = 3 respectively. Fourth and seventh columns represent the respective relative error −RN(z)/I2(z). Fifth and last columns represent the respective relative error bounds given by eqs. (17) or (18).

(9)

Table 2 (Arg(z) = π/4)

First Relative Relative Second Relative Relative

|z| I2(z) order approx. error error bound order approx. error error bound

5 -3.171562e-7 + .00571 .0159 -3.174603e-7 + .000125 .000539

.000055554i .000055555i .000055555i

10 -1.976706e-8 + .00144 .00394 -1.984126e-8 + 7.14e-6 .000023

.000013886i .000013888i .000013888i

20 -1.240075e-9 + .000357 .000984 -1.240079e-9 + 4.46e-7 1.43e-6

3.47222e-6 i 3.472222e-6 i 3.47777 e-6 i

50 -3.1746029e-11 + .0000571 .000157 -3.1746031e-11 + 1.14e-8 3.67e-8 5.555555492e-7i 5.555555555e-7i 5.555555555e-7i

100 -1.984126973e-12 1.14e-8 .0000394 -1.984126984e-12 7.14e-10 2.3e-9

+1.388888887e-7i 1.38888888e-7i +1.38888888e-7i

Table 3 (Arg(z) = π/2)

First Relative Second Relative

|z| I2(z) order approx. error order approx. error 5 .000056023 .000055555 .0111 .000055873 .000119 10 .000013908 .000013888 .00143 .000013908 7.21e-6 20 3.473463e-6 3.472222e-6 .000357 3.473462e-6 4.47e-7 50 5.555873079e-7 5.55555555e-7 .0000572 5.555873016e-7 1.14e-8 100 1.3889087311e-7 1.38888888e-7 .0000143 1.3889087301e-7 7.14e-10

Table 4 (Arg(z) = 3π/4)

First Relative Second Relative

|z| I2(z) order approx. error order approx. error

5 -3.171930e-7- .00571 -3.174603e-7- .000114

.000055549 i -.000055555 i .000055555 i

10 -1.984021e-8 - .00143 -1.984126e-8 - 7.14e-6

.00001388879 i -.00001388889 i .00001388889 i

20 -1.240075e-9- .000357 -1.240079e-9 - 4.46e-7

3.472220e-6 i -3.472222e-6 i 3.472222e-6 i

50 -3.1746029e-11- .0000571 -3.17460317e-11- 1.14e-8 5.5555554e-7 i -5.5555555e-7 i 5.5555555e-7 i

100 -1.984126973e-12- .0000143 -1.984126984e-12- 7.14e-10 1.388888888e-7 i -1.388888889e-7 i 1.388888889e-7 i

Figure 3. Second, third and fifth columns represent the integral I2(z) = log G(z +1)−14z2−z log Γ(z + 1) + 12z2+12z +121 log z + log A, approximation (10) for N = 2 and approximation (10) for N = 3 respectively. Fourth and sixth columns represent the respective relative error −RN(z)/I2(z).

4. Conclusions

The coefficients of the analytic expansion at z = 0 of the function in the right hand side of (2) (which involve the logarithm of the double gamma function G(z)) are given in terms of the zeta function. After introducing an integral representation of the zeta function in this formula, we have derived the integral representation (8) for the function given in the right hand side of (2). By analytical continuation, this integral representation is valid for |Arg(z)| < π. Introducing the expansion of the integrand (9) into this integral, we have obtained the asymptotic expansion (12) of log G(z + 1) uniformly valid for

|Arg(z)| ≤ π − δ < π. The expansion (9) verifies the error test for real 0 ≤ x < 2π and then, an accurate error bound (17) have been obtained for the remainder in the expansion (12) for real z. The bound (17) is not satisfactory for Arg(z) close to ±π/2 and not valid for π/2 ≤ |Arg(z)| < π. Using the freedom we have when choosing the parameter ϕ in (11), a more accurate error bound has been obtained for Arg(z) close to ±π/2. Moreover, this bound is valid for the whole sector |Arg(z)| < π. Numerical experiments

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in table 2 show the accuracy of the expansion (12) for real or complex z and the bounds (17) and (18) for real z. Error bounds for complex z are less realistic.

5. Acknowledgments

This work was stimulated by conversations with Richard Askey and Javier Sesma. The continued interest of Javier Sesma has been quite encouraging. We are grateful to the anonymous referees for the careful examination of the paper and the improving suggestions. The financial support of Comisi´on Interministerial de Ciencia y Tecnologia is acknowledged.

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References

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[3] E. W. Barnes, Genesis of the double gamma function, Proc. London Math. Soc., 31 (1900) 358-381.

[4] E. W. Barnes, The theory of the double gamma function, Philos. Trans. Roy. Soc. London Ser A, 196 (1901) 265-388.

[5] E. L. Basor and P. J. Forrester, Formulas for the evaluation of Toeplitz determinants with rational generating functions, Math. Nachr., 170 (1994) 5-18.

[6] J. Billingham and A. C. King, Uniform asymptotic expansions for the Barnes double gamma function, Proc. Roy. Soc. London Ser A, 453 (1997) 1817-1829.

[7] P Cassou-Nogu´es, Analogues p-adiques des fonctions Γ-multiples, Journ´ees Arithm´etiques de Lu- miny (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), pp. 43-55, Ast´erisque, 61 Soc.

Math. France, Paris, 1979.

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